That's a red herring honestly, because you can't actually do much with that entity until you get a tax ID (from the IRS) and a bank account (where KYC is very tight). The point being that from a tax standpoint these "anonymous" companies are pretty useless. They can however obfuscate ownership of assets.

Example: sure you can transfer your house into South Dakota LLC, and yes when someone looks up the ownership in records, they won't be able to see who owns that house, but there is no tax being avoided.

How's it work if you own the house (purchased by a transfer from ??? to your lawyer) and then sell it (funds sent to your lawyer and then to ???) ?

Probably couldn't get a US mortgage without that tax ID or bank account.

If you rent it out, I guess the contract should be with the LLC and then you run into the tax ID/LLC issues, but people will probably sign a lease with whomever and pay whomever they're told (whether it's the LLC or not).

It depends on what tax you’re theoretically trying to avoid (Your definitely not avoiding property taxes). if you don’t pay tax on the rental income, or capital gains your breaking the law, the question is if the auditors show up, are you really going to be safe? And the answer is possibly? But it would take a ton of work and breaking a whole bunch of other laws along the way (offshore banking relationships in non-cooperation countries, mail drops, repatriation of assets problems etc) , and at the end of the day the government will just put a lien on the house until they’ve gotten what they want.